Related papers: Classifying complements for groups. Applications

Let $\phi: G \rightarrow H$ be a group homomorphism such that $H$ is a totally disconnected locally compact (t.d.l.c.) group and the image of $\phi$ is dense. We show that all such homomorphisms arise as completions of $G$ with respect to…

Let $G = H\times A$ be a group, where $H$ is a purely non-abelian subgroup of $G$ and $A$ is a non-trivial abelian factor of $G$. Then, for $n \geq 2$, we show that there exists an isomorphism $\phi : Aut_{Z(G)}^{\gamma_{n}(G)}(G)…

A group homomorphism eta:A-> H is called a localization of A if every homomorphism phi:A-> H can be `extended uniquely' to a homomorphism Phi:H-> H in the sense that Phi eta = phi. This categorical concepts, obviously not depending on the…

We make explicit some conditions on a semi-abelian category D such that, for any abelian group A in D and any object Y in D, the cohomology group homomorphisms with coefficients in A, induced by the inclusion of the abelian objects of D at…

Isomorphism is central to the structure of mathematics and has been formalized in various ways within dependent type theory. All previous treatments have done this by replacing quantification over sets with quantification over groupoids of…

Given a finite group $G$ with identity $e$ and a normal subgroup $H$ of $G$, the subgroup sum graph $\Gamma_{G,H}$ (resp. extended subgroup sum graph $\Gamma_{G,H}^+$) of $G$ with respect to $H$ is the graph with vertex set $G$, in which…

Let $G$ be a topological group and $A$ a topological $G$-module (not necessarily abelian). In this paper, we define $H^{0}(G,A)$ and $H^{1}(G,A)$ and will find a six terms exact cohomology sequence involving $H^{0}$ and $H^{1}$. We will…

Let $H: V(G) \rightarrow 2^{\mathbb{N}}$ be a set mapping for a graph $G$. Given a spanning subgraph $F$ of $G$, $F$ is called a {\it general factor} or an $H$-{\it factor} of $G$ if $d_{F}(x)\in H(x)$ for every vertex $x\in V(G)$.…

We derive a formula connecting the orders of the automorphism groups of a finite group and of its covering groups.

Every Lie group $G$ carries a distinguished algebra of particularly well-behaved real-analytic mappings: The entire functions $\mathcal{E}(G)$. They were introduced for the purposes of strict deformation quantization. This paper establishes…

This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative…

Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an…

We define the completion of an associative algebra $A$ in a set $M=\{M_1,\dots,M_r\}$ of $r$ right $A$-modules in such a way that if $\mathfrak a\subseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the (right) module…

We give new characterizations of sofic groups: -- A group $G$ is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups. -- A group $G$ is sofic if and only if any system of equations…

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all…

Let $\mathcal C$ be category over a commutative ring $k$, its Hochschild-Mitchell homology and cohomology are denoted respectively $HH_*(\mathcal C)$ and $HH^*(\mathcal C).$ Let $G$ be a group acting on $\mathcal C$, and $\mathcal C[G]$ be…

In this note, starting with any group homomorphism $f\colon\Gamma\to G$, which is surjective upon abelianization, we construct a universal central extension $u\colon U\twoheadrightarrow G,$ UNDER $\Gamma$ with the same surjective property,…

Let $G$ and $H$ be two simple graphs. A bijection $\phi:V(G)\rightarrow V(H)$ is called an isomorphism between $G$ and $H$ if $(\phi v_i)(\phi v_j)\in E(H)$ $\Leftrightarrow$ $v_i v_j\in E(G)$, $\forall v_i,v_j \in V(G)$. In the case that…

Assume $G$ is a polycyclic group and $\phi:G\to G$ an endomorphism. Let $G\ast_{\phi}$ be the ascending HNN extension of $G$ with respect to $\phi$; that is, $G\ast_{\phi}$ is given by the presentation $$G\ast_{\phi}= < G, t \ |\ t^{-1}gt =…

The homological properties of localizations and completions of metabelian groups are studied. It is shown that, for $R=\mathbb Q$ or $R=\mathbb Z/n$ and a finitely presented metabelian group $G$, the natural map from $G$ to its…